常用数列求和公式

2012-7-19 foooy other

1. 1+2+3+......+n=n(n+1)/2
2. 1^2+2^2+3^2+......+n^2=n(n+1)(2n+1)/6
3. 1^3+2^3+3^3+......+n^3=( 1+2+3+......+n)^2=n^2*(n+1)^2/4
4. 1*2+2*3+3*4+......+n(n+1)=n(n+1)(n+2)/3
5. 1*2*3+2*3*4+3*4*5+......+n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4

6. 1+3+6+10+15+......
=1+(1+2)+(1+2+3)+(1+2+3+4)+......+(1+2+3+...+n)
=[1*2+2*3+3*4+......+n(n+1)]/2
=n(n+1)(n+2)/6

7. 1+2+4+7+11+......+ n
=1+(1+1)+(1+1+2)+(1+1+2+3)+......+(1+1+2+3+...+n)
=(n+1)*1+[1*2+2*3+3*4+......+n(n+1)]/2
=(n+1)+n(n+1)(n+2)/6

8. 1/2+1/2*3+1/3*4+......+1/n(n+1)
=1-1/(n+1)=n/(n+1)

9. 1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+......+1/(1+2+3+...+n)
= 2/2*3+2/3*4+2/4*5+......+2/n(n+1)=(n-1)/(n+1)
10. 1/1*2+2/2*3+3/2*3*4+......+(n-1)/2*3*4*...*n
=(2*3*4*...*n-1)/2*3*4*...*n

11. 1^2+3^2+5^2+..........(2n-1)^2=n(4n^2-1)/3
12. 1^3+3^3+5^3+..........(2n-1)^3=n^2(2n^2-1)
13. 1^4+2^4+3^4+..........+n^4=n(n+1)(2n+1)(3n^2+3n-1)/30
14. 1^5+2^5+3^5+..........+n^5=n^2 (n+1)^2 (2n^2+2n-1) /12

15. 1+2+2^2+2^3+......+2^n=2^(n+1) – 1

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